3.1104 \(\int \frac{c+d x^2}{(e x)^{13/2} \left (a+b x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=182 \[ \frac{8 b^{5/2} (e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (10 b c-11 a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{77 a^{7/2} e^8 \left (a+b x^2\right )^{3/4}}-\frac{4 b \sqrt [4]{a+b x^2} (10 b c-11 a d)}{77 a^3 e^5 (e x)^{3/2}}+\frac{2 \sqrt [4]{a+b x^2} (10 b c-11 a d)}{77 a^2 e^3 (e x)^{7/2}}-\frac{2 c \sqrt [4]{a+b x^2}}{11 a e (e x)^{11/2}} \]

[Out]

(-2*c*(a + b*x^2)^(1/4))/(11*a*e*(e*x)^(11/2)) + (2*(10*b*c - 11*a*d)*(a + b*x^2
)^(1/4))/(77*a^2*e^3*(e*x)^(7/2)) - (4*b*(10*b*c - 11*a*d)*(a + b*x^2)^(1/4))/(7
7*a^3*e^5*(e*x)^(3/2)) + (8*b^(5/2)*(10*b*c - 11*a*d)*(1 + a/(b*x^2))^(3/4)*(e*x
)^(3/2)*EllipticF[ArcCot[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(77*a^(7/2)*e^8*(a + b*x^2)
^(3/4))

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Rubi [A]  time = 0.402303, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{8 b^{5/2} (e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (10 b c-11 a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{77 a^{7/2} e^8 \left (a+b x^2\right )^{3/4}}-\frac{4 b \sqrt [4]{a+b x^2} (10 b c-11 a d)}{77 a^3 e^5 (e x)^{3/2}}+\frac{2 \sqrt [4]{a+b x^2} (10 b c-11 a d)}{77 a^2 e^3 (e x)^{7/2}}-\frac{2 c \sqrt [4]{a+b x^2}}{11 a e (e x)^{11/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)/((e*x)^(13/2)*(a + b*x^2)^(3/4)),x]

[Out]

(-2*c*(a + b*x^2)^(1/4))/(11*a*e*(e*x)^(11/2)) + (2*(10*b*c - 11*a*d)*(a + b*x^2
)^(1/4))/(77*a^2*e^3*(e*x)^(7/2)) - (4*b*(10*b*c - 11*a*d)*(a + b*x^2)^(1/4))/(7
7*a^3*e^5*(e*x)^(3/2)) + (8*b^(5/2)*(10*b*c - 11*a*d)*(1 + a/(b*x^2))^(3/4)*(e*x
)^(3/2)*EllipticF[ArcCot[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(77*a^(7/2)*e^8*(a + b*x^2)
^(3/4))

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Rubi in Sympy [A]  time = 43.2482, size = 173, normalized size = 0.95 \[ - \frac{2 c \sqrt [4]{a + b x^{2}}}{11 a e \left (e x\right )^{\frac{11}{2}}} - \frac{2 \sqrt [4]{a + b x^{2}} \left (11 a d - 10 b c\right )}{77 a^{2} e^{3} \left (e x\right )^{\frac{7}{2}}} + \frac{4 b \sqrt [4]{a + b x^{2}} \left (11 a d - 10 b c\right )}{77 a^{3} e^{5} \left (e x\right )^{\frac{3}{2}}} - \frac{8 b^{\frac{5}{2}} \left (e x\right )^{\frac{3}{2}} \left (11 a d - 10 b c\right ) \left (\frac{a}{b x^{2}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2}\middle | 2\right )}{77 a^{\frac{7}{2}} e^{8} \left (a + b x^{2}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)/(e*x)**(13/2)/(b*x**2+a)**(3/4),x)

[Out]

-2*c*(a + b*x**2)**(1/4)/(11*a*e*(e*x)**(11/2)) - 2*(a + b*x**2)**(1/4)*(11*a*d
- 10*b*c)/(77*a**2*e**3*(e*x)**(7/2)) + 4*b*(a + b*x**2)**(1/4)*(11*a*d - 10*b*c
)/(77*a**3*e**5*(e*x)**(3/2)) - 8*b**(5/2)*(e*x)**(3/2)*(11*a*d - 10*b*c)*(a/(b*
x**2) + 1)**(3/4)*elliptic_f(atan(sqrt(a)/(sqrt(b)*x))/2, 2)/(77*a**(7/2)*e**8*(
a + b*x**2)**(3/4))

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Mathematica [C]  time = 0.218361, size = 132, normalized size = 0.73 \[ \frac{\sqrt{e x} \left (8 b^2 x^6 \left (\frac{b x^2}{a}+1\right )^{3/4} (11 a d-10 b c) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right )-2 \left (a+b x^2\right ) \left (a^2 \left (7 c+11 d x^2\right )-2 a b x^2 \left (5 c+11 d x^2\right )+20 b^2 c x^4\right )\right )}{77 a^3 e^7 x^6 \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)/((e*x)^(13/2)*(a + b*x^2)^(3/4)),x]

[Out]

(Sqrt[e*x]*(-2*(a + b*x^2)*(20*b^2*c*x^4 - 2*a*b*x^2*(5*c + 11*d*x^2) + a^2*(7*c
+ 11*d*x^2)) + 8*b^2*(-10*b*c + 11*a*d)*x^6*(1 + (b*x^2)/a)^(3/4)*Hypergeometri
c2F1[1/4, 3/4, 5/4, -((b*x^2)/a)]))/(77*a^3*e^7*x^6*(a + b*x^2)^(3/4))

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Maple [F]  time = 0.062, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{-{\frac{13}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/(e*x)^(13/2)/(b*x^2+a)^(3/4),x)

[Out]

int((d*x^2+c)/(e*x)^(13/2)/(b*x^2+a)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \left (e x\right )^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(13/2)),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(13/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x} e^{6} x^{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(13/2)),x, algorithm="fricas")

[Out]

integral((d*x^2 + c)/((b*x^2 + a)^(3/4)*sqrt(e*x)*e^6*x^6), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**2+c)/(e*x)**(13/2)/(b*x**2+a)**(3/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \left (e x\right )^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(13/2)),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(13/2)), x)